Statistics and Probability

In probability theory, a trial refers to a single occurrence or performance of a random experiment. The result of a trial is called an outcome. When each outcome in a sample space is equally probable, they are termed as equally likely outcomes. The relative frequency of an event is the ratio of the number of times the event occurs to the total number of trials. The set of all possible outcomes of an experiment is termed the sample space, denoted by \( U \). An event is a subset of the sample space, representing a specific set of outcomes. The probability of an event \( A \) is given by \( P(A) = \frac{n(A)}{n(U)} \), where \( n(A) \) is the number of outcomes in event \( A \) and \( n(U) \) is the total number of outcomes in the sample space. Complementary events refer to events that are mutually exclusive to each other. If \( A \) is an event, then \( A' \) (or not \( A \)) is the event that \( A \) does not occur. The expected number of occurrences of an event is the number of times we expect the event to occur in a given number of trials, based on its probability.

Key Formulae:

$$ P(A) = \frac{n(A)}{n(U)} $$

$$ P(A') = 1 - P(A) $$

$$ \text{Expected number of occurrences} = \text{probability of event} \times \text{number of trials} $$

Example:

Consider a fair six-sided die. The sample space \( U \) is \( \{1, 2, 3, 4, 5, 6\} \). Let event \( A \) be the event that the die shows an even number. So, \( A = \{2, 4, 6\} \).
Using the formula, the probability of event \( A \) is:

$$ P(A) = \frac{n(A)}{n(U)} = \frac{3}{6} = \frac{1}{2} $$

Thus, if we roll the die 100 times, the expected number of times we get an even number is:

$$ \text{Expected number of occurrences} = \frac{1}{2} \times 100 = 50 $$